Overview Definitions
Sparsity and Meagerness Bounds Show these bounds very loose Define Informational Meagerness
Based on Informational Dominance Show that it can be slightly loose
Concurrent Rate
Source i desires communication rate di. Rate r is achievable if rate vector
[ rd1, rd2, …, rdk ] is achievable Rate region interval of R+
Def: “Network coding rate” (or NCR) := sup { r : r is achievable }
Upper bounds on rate
[Classical]: Sparsity bound for multicommodity flows
[CT91]: General bound for multi-commodity information networks
[B02]: Application of CT91 to directed network coding instances; equivalent to sparsity.
[KS03]: Bound for undirected networks with arbitrarytwo-way channels
[HKL04]: Meagerness
[SYC03], [HKL05]: LP bound
[KS05]: Bound based on iterative d-separation
Vertex-Sparsity
Def: For U V,
VS (G) := minUV VS (U)
Claim: NCR VS (G)
Capacity of edges crossing between U and U
Demand of commodities separated by UVS
(U) :=
Edge-Sparsity Def: For A E,
ES (G) = minAE ES (A)
Claim: Max-Flow ES (G)
But: Sometimes NCR > ES (G)
Capacity of edges in A
Demand of commodities separated in G\AES (A) :=
NCR > Edge-Sparsity
S(1) S(2)
T(2) T(1)
Cut {e} separates S(1) and S(2)
ES ({e}) = 1/2 But rate 1 achievable!
e
Meagerness Def: For A E and P [k],
A isolates P if for all i,j P,S(i) and T(j) disconnected in G\A.
M (G) := minAE M (A)
Claim: NCR M (G)
Capacity of edges in A
Demand of commodities in PM (A) := minP isolated by A
Meagerness & Vtx-Sparsity are weak
Thm: M (Gn) = VS (Gn) = (1),but NCR 1/n.
S(3) S(2)S(n) S(n-1) f2fn-1 f3 S(1)f1
T(1)T(n-1)T(n) T(3)hn-1 h1h3 T(2)h2
g2g3 g1gn-1gn
Gn :=
A Proof Tool
Def: Let A,B E. B is downstream of Aif B disconnected from sources in G\A.Notation: A B.
Claim: If A B then H(A) H(A,B).
Pf: Because S A B form Markov chain.
Proof:
{gn} {gn,T(1),h1}
S(3) S(2)S(n) S(n-1) f2fn-1 f3 S(1)f1
T(1)T(n-1)T(n) T(3)hn-1 h1h3 T(2)h2
g2g3 g1gn-1gn
Gn :=
Lemma: NCR 1/n
Proof:
{gn} {gn,T(1),h1} {S(1),f1,g1,h1}
S(3) S(2)S(n) S(n-1) f2fn-1 f3 S(1)f1
T(1)T(n-1)T(n) T(3)hn-1 h1h3 T(2)h2
g2g3 g1gn-1gn
Gn :=
Lemma: NCR 1/n
Proof:
{gn} {gn,T(1),h1} {S(1),f1,g1,h1}
{S(1),f1,T(2),h2}
S(3) S(2)S(n) S(n-1) f2fn-1 f3 S(1)f1
T(1)T(n-1)T(n) T(3)hn-1 h1h3 T(2)h2
g2g3 g1gn-1gn
Gn :=
Lemma: NCR 1/n
Proof:
{gn} {gn,T(1),h1} {S(1),f1,g1,h1}
{S(1),f1,T(2),h2} {S(1),S(2),f2,g2,h2}
S(3) S(2)S(n) S(n-1) f2fn-1 f3 S(1)f1
T(1)T(n-1)T(n) T(3)hn-1 h1h3 T(2)h2
g2g3 g1gn-1gn
Gn :=
Lemma: NCR 1/n
h3
Proof:
{gn} {gn,T(1),h1} {S(1),f1,g1,h1}
{S(1),f1,T(2),h2} {S(1),S(2),f2,g2,h2}
{S(1),S(2),f2,T(3),h3}
S(3) S(2)S(n) S(n-1) f2fn-1 f3 S(1)f1
T(1)T(n-1)T(n) T(3)hn-1 h1T(2)h2
g2g3 g1gn-1gn
Gn :=
Lemma: NCR 1/n
Proof:
{gn} … {S(1),S(2),…,S(n)}
Thus 1 H(gn) H(S(1),…,S(n)) = n∙r
So 1/n r
S(3) S(2)S(n) S(n-1) f2fn-1 f3 S(1)f1
T(1)T(n-1)T(n) T(3)hn-1 h1h3 T(2)h2
g2g3 g1gn-1gn
Gn :=
Lemma: NCR 1/n
Towards a stronger bound Our focus: cut-based bounds
Given A E, we want to infer thatH(A) H(A,P) where P{S(1),…,S(k)}
Meagerness uses Markovicity:(sources in P) A (sinks in P)
Markovicity sometimes not enough…
Informational Dominance
Def: A dominates B if information in A determines information in Bin every network coding solution.Denoted A B.
Trivially implies H(A) H(A,B) How to determine if A dominates B?
[HKL05] give combinatorial characterization and efficient algorithm to test if A dominates B.
i
Informational Meagerness Def: For A E and P {S(1),…,S(k)},
A informationally isolates P ifAP P.
iM (A) = minP
for P informationally isolated by A
iM (G) = minA E iM (A)
Claim: NCR iM (G).
iCapacity of edges in A
Demand of commodities in P
iMeagerness Example
“Obviously” NCR = 1. But no two edges disconnect t1 and t2 from both
sources!
s1 s2
t1
t2
Informational Dominance Examples1 s2
t1
t2
Our characterization shows A {t1,t2}
H(A) H(t1,t2) and iM (G) = 1
Cut A
i
A bad example: Hn
Thm: iMeagerness gap of Hn is (log |V|)
s(00)s(0)
s(01) s(10) s(11)s(1)
s(ε)
q(00) q(01) q(10) q(11)r(00) r(01) r(10) r(11)
t(00)t(0)
t(01) t(10) t(11)t(1)
t(ε)
Capacity 2-nH2
s(00)s(0)
s(01) s(10) s(11)s(1)
s(ε)
Tn = Binary tree of depth n
Source S(i) iTn
Sink T(i) iTn
t(00)t(0)
t(01) t(10) t(11)t(1)
t(ε)
r(00) r(01) r(10) r(11)
s(00)s(0)
s(01) s(10) s(11)s(1)
s(ε)
t(00)t(0)
t(01) t(10) t(11)t(1)
t(ε)
q(00) q(01) q(10) q(11)
Nodes q(i) and r(i) for every leaf i of Tn
r(00) r(01) r(10) r(11)
s(00)s(0)
s(01) s(10) s(11)s(1)
s(ε)
t(00)t(0)
t(01) t(10) t(11)t(1)
t(ε)
q(00) q(01) q(10) q(11)
Complete bip. graph between sources and q’s
r(00) r(01) r(10) r(11)
s(00)s(0)
s(01) s(10) s(11)s(1)
s(ε)
t(00)t(0)
t(01) t(10) t(11)t(1)
t(ε)
q(00) q(01) q(10) q(11)
(r(a),t(b)) if b ancestor of a in Tn
r(00) r(01) r(10) r(11)
s(00)s(0)
s(01) s(10) s(11)s(1)
s(ε)
q(00) q(01) q(10) q(11)
t(00)t(0)
t(01) t(10) t(11)t(1)
t(ε)
(s(a),t(b)) if a and b cousins in Tn
r(00) r(01) r(10) r(11)
s(00)s(0)
s(01) s(10) s(11)s(1)
s(ε)
q(00) q(01) q(10) q(11)
t(00)t(0)
t(01) t(10) t(11)t(1)
t(ε)
Capacity 2-n
All edges have capacity except (q(i),r(i))
r(00) r(01) r(10) r(11)
s(00)s(0)
s(01) s(10) s(11)s(1)
s(ε)
q(00) q(01) q(10) q(11)
t(00)t(0)
t(01) t(10) t(11)t(1)
t(ε)
Capacity 2-n
Demand of source at depth i is 2-i
Properties of Hn
Lemma: iM (Hn) = (1)
Lemma: NCR < 1/n
Corollary: iMeagerness gap is n=O(log |V|)
We will prove this
Entropy moneybags i.e., sets of RVs
Entropy investments Buying sources and edges, putting into moneybag Loans may be necessary
Profit Via Downstreamness or Info. Dominance Earn new sources or edges for moneybag
Corporate mergers Via Submodularity New Investment Opportunities and Debt Consolidation
Debt repayment
Proof Ingredients
Submodularity of Entropy
Claim: Let A and B be sets of RVs.Then H(A)+H(B) H(AB)+H(AB)
Pf: Equivalent to I( X; Y | Z ) 0.
Proof: Two entropy moneybags:
F(a) = { S(b) : b not an ancestor of a }E(a) = F(a) { (q(b),r(b)) : b is descendant of a }
Lemma: NCR < 1/n
Entropy Investment
Let a be a leaf of Tn
Take a loan and buy E(a).
r(00) r(01) r(10) r(11)
s(00)s(0)
s(01) s(10) s(11)s(1)
s(ε)
q(00) q(01) q(10) q(11)
a
t(00)
Earning Profit
Claim: E(a) T(a)
Pf: Cousin-edges not from ancestors.Vertex r(00) blocked by E(a).
r(00) r(01) r(10) r(11)
s(00)s(0)
s(01) s(10) s(11)s(1)
s(ε)
q(00) q(01) q(10) q(11)
a
Earning Profit
Claim: E(a) T(a)
Result:E(a) gives free upgrade to E(a){S(a)}.Profit = S(a).
r(00) r(01) r(10) r(11)
s(00)s(0)
s(01) s(10) s(11)s(1)
s(ε)
q(00) q(01) q(10) q(11)
a
t(00)
q(00) q(01)
r(00) r(01) r(10) r(11)
s(00)s(0)
s(01) s(10) s(11)s(1)
s(ε)
q(00) q(01) q(10) q(11)
E(aL){S(aL)}
r(00) r(01) r(10) r(11)
s(00)s(0)
s(01) s(10) s(11)s(1)
s(ε)
q(10) q(11)
E(aR){S(aR)}
aL
aR
q(00) q(01)
r(00) r(01) r(10) r(11)
s(00)s(0)
s(01) s(10) s(11)s(1)
s(ε)
q(00) q(01) q(10) q(11)
(E(aL){S(aL)})
(E(aR){S(aR)})
r(00) r(01) r(10) r(11)
s(00)s(0)
s(01) s(10) s(11)s(1)
s(ε)
q(10) q(11)
(E(aL){S(aL)})
(E(aR){S(aR)})
Applying submodularity
r(00) r(01) r(10) r(11)
s(00)s(0)
s(01) s(10) s(11)s(1)
s(ε)
q(00) q(01) q(10) q(11)
(E(aL){S(aL)})
(E(aR){S(aR)})
New Investment
Union term has more edges
Can use downstreamnessor informational dominance again!
(E(aL){S(aL)}) (E(aR){S(aR)}) = E(a)
a
Debt Consolidation Intersection term has only sources
Cannot earn new profit.
Used for later “debt repayment” (E(aL){S(aL)}) (E(aR){S(aR)}) = F(a)
q(00) q(01)r(00) r(01) r(10) r(11)
s(00)s(0)
s(01) s(10) s(11)s(1)
s(ε)
q(10) q(11)
(E(aL){S(aL)})
(E(aR){S(aR)})
a
What have we shown? Let aL,aR be sibling leaves; a is their parent.
H(E(aL)) + H(E(aR)) H(E(a)) + H(F(a)) Iterate and sum over all nodes in tree
where r is the root. Note: E(v) = F(v) {(q(v),r(v))} when v is a leaf
vl
vFHrEHlEH nonleaf leaf
))(())(())((
v
l l
vFHrEH
HlFH lrlq
nonleaf
leaf leaf
))(())((
)())(( ))(),((
Debt Repayment
Claim:
Pf: Simple counting argument.
vl
vFHlFH nonleaf leaf
))(())((
v
l l
vFHrEH
HlFH lrlq
nonleaf
leaf leaf
))(())((
)())(( ))(),((
))(()( leaf
))(),(( rEHHl
lrlq